3.1 A smooth sine wave scenario

Given a series $\{y_t\}_{t=1}^{t^{max}}$ , HRF specifies the trend value $y^\star _t$ in period $t$ as linear regression forecast made $h$ periods earlier based on a small number of lags $p$ up to $y_{t-h}$ . The “cycle” $\{c_t\}_{t=h+p}^{t^{max}}$ is then defined as the prediction error. Regression coefficients homogeneous across all periods $t$ are obtained from a preceding OLS estimation over the entire sample. Regarding $h$ , Hamilton (Reference Hamilton2018) recommends a value corresponding to roughly a quarter of a typical business cycle. Specifically, for quarterly data, he fixes $h=8$ and $p=4$ . The regression reads

\begin{align*} y_t = {\beta }_o \, + \, \sum _{k=1}^p \, {\beta }_k \, y_{t-h-k+1} \, + \, u_t \,, \qquad t = h+p, \, \ldots \,, t^{max} \end{align*}

and with the resulting estimates $\widehat{\beta }_0, \widehat{\beta }_1, \, \ldots \,, \widehat{\beta }_p$ , the HRF trend $y_t^{\star }$ and the cyclical component $c_t$ in period $t$ are given by

(1) \begin{eqnarray} \begin{array}{rcl} y_t^{\star } &=& y_t^{\star,\textrm{HRF}} = \displaystyle \widehat{\beta }_o \, + \, \sum _{k=1}^p \, \widehat{\beta }_k \, y_{t-h-k+1} \,, \qquad t = h+p, \, \ldots \,, T \\ c_t &=& c_t^{\textrm{HRF}} = y_t \, - \, y^\star _t \, \, = \, \, \widehat{u}_t \end{array} \end{eqnarray}

Figure 1. Application of HRF in a sine wave scenario.

To illustrate HRF limitations, consider the examples depicted in Figure 1. The upper-left panel plots a sine wave with a period of $T = 9.50$ years around a straight trend line $y^\star _t$ with a slope of 3% per year. The series to be examined is thus

(2) \begin{align} y_t = y^\star _t \, + \, c_t. \end{align}

The bold red solid line in the upper-left panel represents the estimated HRF trend, which covers the data most of the time. Excluding the spikes in the middle of the sample, the lower-left panel illustrates the significantly lower amplitude of the cyclical component derived from HRF vis-à-vis the sine wave oscillations (the dotted line); the proportion is approximately 1:5. Importantly, a quarter-cycle phase shift between the two series further indicates that HRF might lead to a significant misjudgment in accurately timing the true underlying business cycles.Footnote ⁵ For a similar point, for applied work, see Cauvel (Reference Cauvel2022).

To demonstrate another potentially spurious outcome of HRF between years 20 and 30, we introduce a recession that is twice as deep as that in the other cycles. While HRF predictions $y_t^\star$ reflect this change, the corresponding downward revisions significantly overshoot the true contraction. Consequently, the cyclical component exhibits the first positive spike. Two subsequent spikes, negative and positive, occur when the sine wave oscillations return to their original amplitude. Thus, HRF suggests a sudden and extraordinary recovery just in the deepest recession and a sizeable contraction in the following recovery, around $t = 30$ .Footnote ⁶

The lower-left panel of Figure 1 also contrasts the HRF cyclical component with that from the ordinary HP filter (the thinner blue solid line), which succeeds almost perfectly in tracing out the regular pattern of the sine wave oscillations; the correlation is as high as 0.975. Conversely, the HP filter tends to incorporate too much of the cycle into the trend, resulting in an estimated cyclical component with only three-fifths of the true cycle’s amplitude.

3.2 The role of stochastic noise

The example of a deterministic sine wave shows that HRF seems unsuitable for filtering smoothly evolving time series. It is useful to add stochastic noise to this example to assess the potential improvements the filter may offer.

The deterministic oscillations in (2) with an amplitude $\pm \bar{\alpha } = 0.05$ are disturbed by random draws from a normal distribution with a standard deviation related to $\bar{\alpha }$ by a factor $m$ . For $-2 \leq t \leq 55$ , the cyclical component $c_t$ is given by

(3) \begin{eqnarray} \begin{array}{rcl} c_t &=&{\alpha }_t \, \sin (\omega \,t) \, + \,{\varepsilon }_t \,, \qquad{\alpha }_t = \left \{ \begin{array}{ll} 2\,\bar{\alpha } & 23.75 \leq t \leq 28.50 \\ \bar{\alpha } = 0.05 & \textrm{else} \end{array} \right . \\ & & \omega = 2\pi/9.50 \,, \quad{\varepsilon }_t \sim N(0,\sigma ^2) \,, \quad \sigma = m \, \bar{\alpha } \end{array} \end{eqnarray}

Simulations in the two panels on the right-hand side of Figure 1 are based on $m = 0.15$ . The slightly perturbed cycle $c_t$ is the black dotted line in the lower-right panel. What is particularly noticeable is the substantial, overproportional noise in the cyclical component $c_t^{\textrm{HRF}}$ indicated by HRF-CC, the bold red line. Moreover, its cycle variability not only increases compared to that of the deterministic case but is now of a similar magnitude to that of the true cycle. In this regard, HRF is better than the HP filter (blue), whose cyclical component exhibits only minimal alterations compared to the deterministic case. However, considerable doubts arise when attempting to identify a general pattern in these fluctuations. It appears that the contractions and expansions of $c_t^{\textrm{HRF}}$ end considerably before those of the true cycle. In essence, the phase shift highlighted in the lower-left panel seems to persist in the stochastic environment. Contemporaneous and lagged cross-correlations support this visual impression: a disappointingly low Corr $(c_t,c_{t}^{\textrm{HRF}}) = 0.237$ , whereas Corr $(c_t,c_{t-k}^{\textrm{HRF}}) = 0.828$ for lag $k = h = 8$ (compare to a large Corr $(c_t,c_t^{\textrm{HP}}) = 0.964$ for the HP filter).

Table 1. Statistics of the estimated cyclical components $c_t^{\textrm{{HP}}}$ and $c_t^{\textrm{{HRF}}}$ from (3), varying with the noise level $m$

Note: “std.” is the standard deviation of $c_t^{\textrm{{HP}}}$ and $c_t^{\textrm{{HRF}}}$ , respectively, in relation to that of the true cycle $c_t$ . “C( $k$ )” for $k = 0, 5, 6, 7, 8$ quarters stands for the cross correlation coefficient $\text{Corr}(c_t, c^j_{t-k})$ , $j =$ HP, HRF. Boldface figures indicate maximal correlation across all lags.

Table 1 examines the impact of varying noise levels $m$ . The random sequences $\tilde{\varepsilon }_t$ , drawn from the unit normal so that ${\varepsilon }_t = \sigma \, \tilde{\varepsilon }_t = m \, \bar{\alpha }\,\tilde{\varepsilon }_t$ , are uniform across all $m$ . Differences in statistics (i.e., standard deviation and correlation with the true cycle) are thus solely attributed to changes in the noise level. For the HP filter, the statistics remain consistent across noise levels. Although the contemporaneous correlation slightly decreases with higher $m$ , even for elevated levels, the cyclical pattern maintains close alignment with the true pattern. Conversely, the impacts on HRF are more pronounced. Initially, $c_t^{\textrm{ HRF}}$ has insufficient variability at $m = 0$ , aligning more closely with the true cycle as $m$ increases. At higher levels, however, some overshooting is observed. However, in the deterministic case, both contemporaneous and lagged correlations are unsatisfactorily low, and a modest increase in randomness notably improves these statistics. Thus, at high noise levels such as $m = 0.50$ , the contemporaneous correlation becomes a stronger indicator of the connection between the estimated and true cyclical components. Nevertheless, the correlation coefficients with $c_t$ do not surpass those of the HP filter.

These simple experiments provide an initial heuristic to assess the relative merits of HRF and the HP filter. They caution against using HRF when dealing with data showing excessive regularity. Regarding stochastic perturbations, the findings demonstrate that HP is more reliable than HRF in recognizing the true cyclical pattern, even under high noise levels ( $m$ ). However, relying on HP results in a systematic underestimation of the cycle’s overall variability, an aspect seemingly better captured by HRF.

4.1 Specification of stochastic processes

We calibrate the data-generating process to an empirical business cycle variable. We work with the quarterly real gross value added (GVA) of the US nonfinancial corporate business, measured in logarithms, instead of GDP.Footnote ⁷ GVA is preferred over GDP because it aligns closely with the output variable in many small-scale macro models, which typically represents the output of the firm sector alone.

For the remainder of the paper, the underlying time unit will shift to a year from a quarter. The simulation’s time horizon remains constant at 60 years, a standard duration for many post-WWII empirical investigations.

4.1.1 Three growth regimes

In specifying the output trend, two versions will be distinguished, considering a pivotal distinction in economic theory: the concept of a deterministic versus a stochastic trend. The former is typically depicted as an increasing line with a fixed slope, while a random walk with a constant drift represents the latter. However, for the postwar US economy, maintaining the assumption of a constant growth trend over an extended period is unrealistic; indeed, growth rates were consistently declining. A simplified narrative identifies three growth regimes. Drawing on a recent discussion by Hall (Reference Hall2020), these regimes include an era of modernization from 1950 to 1975, an era of liberalism from 1980 to 2000, and subsequently, an era of knowledge-based growth. Associated with the concept of productivity slowdown, growth rates progressively decrease from one regime to the next.

Correspondingly, we postulate three regimes, each spanning 20 years, with growth rates of 5%, 3.5%, and 2%, respectively. The abrupt transitions between regimes are gently smoothed using a moving average around the break dates to mitigate any undue disadvantage to the HP filter. Specifically, we employ the two-sided moving average TS-MA $(x_t,4)$ with equal weights, considering a series extended by 4 quarters on each side of a value $x_t$ . Applying this to the step function representing the three growth rates, we obtain a continuous relationship of (actual or expected) trend growth rates $g^e_t$ in periods $t = 0,\, 0.25,\, 0.50, \, \ldots \,,\, 60.00$ , albeit with kinks during regime changes:

(4) \begin{align} g^e_t = \textrm{TS-MA}(\tilde{g}^e_t,4) \end{align}

(5) \begin{align} \tilde{g}^e_t = \begin{cases}0.050 & \textrm{if } t \lt 20\\ 0.035 & \textrm{if } 20 \leq t \lt 40\\ 0.020 & \textrm{if } t \geq 40\\ \end{cases}\end{align}

Typically, $g^e_t = \tilde{g}^e_t$ . We employ the acronyms DT and ST to denote the deterministic and stochastic trend concepts, respectively. Initialized with a $y^\star _0 = 0$ and variance ${\sigma }^2_{\varepsilon }$ for the random walk, the two trend series are expressed as follows:

\begin{align*} y^\star _t = y^\star _{t-0.25} \, + \, 0.25\, g^e_t \qquad \qquad \textrm{(DT)} \end{align*}

\begin{align*} y^\star _t = y^\star _{t-0.25} \, + \, 0.25\, g^e_t \, + \,{\varepsilon }_t \, \qquad{\varepsilon }_t \sim N(0,{\sigma }^2_{\varepsilon }),\,iid \qquad \qquad \textrm{(ST)} \end{align*}

4.1.2 Specification of the cyclical component

In outlining our methodology for assessing the HRF and the HP filter using artificial data from a stochastic process, our approach aligns with recent works by Hodrick (Reference Hodrick2020, Section 10) and Schüler (2021, Section 4). However, we contend that our experiments are more suitable for studying business cycle data. This implies that we directly impose a desired periodicity on the cyclical component, which contrasts with standard time series analysis. In such analyses, there is often little discussion on whether the resulting cyclical component adequately captures such a fundamental characteristic of the business cycle. Our findings, therefore, complement the investigations conducted by Hodrick (Reference Hodrick2020) and Schüler (Reference Schüler2021).Footnote ⁸

To construct the cyclical component $c_t$ , we adopt a sine wave with suitable random disturbances to ensure the desired spectrogram shape. Two noise sources are considered: one randomly varies the period of a sine wave from one full cycle run to the next, and the other adds a stochastic moving average process to these waves.

We begin by describing the first concept of a regular oscillatory motion, denoted $s_t$ . It features a constant amplitude $\alpha$ and an average period $T$ . The random periods of a single cycle run are drawn from an interval $[T - \Delta T, T + \Delta T]$ with equal probabilities. Denoting the uniform distribution as $U$ , $\phi$ is a parameter that may shift the waves in time, and $t \in \mathbb{R}$ is a point in time. The relationship $s_t = s_t(T, \Delta T, \phi, \alpha )$ is defined as follows:

(6) \begin{eqnarray} s_t(T,\Delta T, \phi,{\alpha }) &=&{\alpha } \, \sin [\,{\omega }_{\,t}\,(t-\tau _{\,t}) + \,\phi \, ] \, \quad \textrm{where:} \nonumber \\{\omega }_{\,t} & = & 2 \pi/T(k) \quad \textrm{for} t_{k-1} \leq \, t \lt \, t_k \nonumber \\ T(j) & \sim & U(T \!- \Delta T, T \!+ \Delta T) \, \qquad j = 1, 2, \, \ldots \\ t_k & = & \sum _{j=1}^k \, T(j) \, \qquad t_0 \, = \, 0 \nonumber \\ \tau _{\,t} &=& t_{k-1} \quad \textrm{for } t_{k-1} \leq \, t \lt \, t_k \nonumber \end{eqnarray}

In a continuous-time scenario with a transient $\phi = 0$ , a cycle initiates at $t = t_{k-1}$ when the argument of the sine function in the first row is ${\omega }_{\,t} (t-t_{k-1}) = 0$ . Note that $t_k - t_{k-1} = T(k)$ . Thus, as $t$ approaches $t_k$ from the left, the argument tends toward ${\omega }_{\,t} (t_k - t_{k-1}) = [2\pi/T(k)] \,\, T(k) = 2\pi$ , and the function converges to $\sin (2\pi ) = 0$ , from whereon the next cycle starts. With a nonzero shift factor $\phi$ , we have $\sin (\phi )$ at these connection points. In the context of specifying the cyclical component $c_t$ below, the function $s_t$ only needs to be evaluated in quarterly intervals at $\,t = 0, \, 0.25, \, 0.50, \, \ldots \, 60$ .

Function (6) prompts the question of selecting a “typical” business cycle period, conventionally considered no longer than 8 years in discussions around Hodrick–Prescott detrending or bandpass filters. Contrary to this perspective, recent research by Beaudry et al. (Reference Beaudry, Galizia and Portier2020) and Barrales-Ruiz and von Arnim (Reference Barrales-Ruiz and von Arnim2021) challenges this notion. Their analysis of spectrograms for cycle-related empirical variables without long-run growth (e.g., working hours per capita or job finding rates) reveals a distinct peak occurring between 38 and 40 quarters. This is also the order of magnitude that guides our investigation.

Motions $s_t$ constitute the core of our cyclical component, but the concern arises that a single motion might exhibit excessive regularity, potentially disadvantaging HRF relative to HP. Thus, we combine two sine waves, each with a unique (expected) period $T$ : one with $T = 9.50$ years and the second with T = 7.00 years but half the amplitude of the former. This choice is motivated by a second minor peak for a shorter period in several of the aforementioned spectrograms.

Turning to our second noise concept, introducing additional random perturbations in these oscillations, a three-lag moving average process, MA(3), proves sufficient for our purpose. It is unnecessary to include an autoregressive part, as sine waves already fulfill their role. Therefore, with respect to the amplitude $\alpha$ , the MA coefficients $\theta _1$ , $\theta _2$ , $\theta _3$ , and the variance of the innovations ${\sigma }^2_{\eta }$ , our specification for the cyclical component $c_t$ is as follows, at times $\,t = 0, \, 0.25, \, 0.50, \, \ldots \, 60:$

(7) \begin{eqnarray} \begin{array}{rcl} c_t &=& s_t(9.50, 0.00, 0,{\alpha }) \, + \, s_t(7.00,\Delta T,0.70\cdot 7.00,{\alpha }/2) \\ & & + \, \,eta_{\,t} \, + \, \sum _{j=1}^3 \, \theta _j \, \eta _{\,t-0.25\,j} \, \quad \eta _t \sim N(0,{\sigma }^2_{\eta }) \, \quad \Delta T = 1.00 \end{array} \end{eqnarray}

A shift $\phi = 0.70 \cdot 7.00$ in the second sine wave in (7) introduces more irregularity in the composite motion than for example, $\phi = 0.50 \cdot 7.00$ . Note that the parameters $\theta _1$ , $\theta _2$ , $\theta _3,{\sigma }^2_{\eta }$ will generally differ when (7) is combined with (DT) and (ST). Overall, the series serves as a representative example of what detrending procedures typically produce in empirical analyses of GDP and similar macroeconomic variables. Therefore, we posit that the series

(8) \begin{eqnarray} y_t &=& y^\star _t \, + \, c_t \, \quad t = 0, \, 0.25, \, 0.50, \, \ldots \, 60 \quad \end{eqnarray}

resulting from the combination of (7) with the trends (DT) and (ST) provided a realistic test bed that HRF and HP should be able to cope with.

4.2 Calibration of the numerical coefficients

Before delving into the details, it is crucial to outline the procedure used to determine the parameters of the data-generation processes, ${\beta } := ({\alpha }, \theta _1, \theta _2, \theta _3, \sigma _\eta, \sigma _{\varepsilon })$ . The design of this procedure and its outcomes are adopted from Franke et al. (2023). It has already been pointed out that independent of the specific values assigned to $\beta$ , the criterion of a realistic periodicity of the simulated series $y_t$ is directly fulfilled by (7) by construction.

Turning to the other features shared with the empirical GVA, we focus on the series of its first differences. Specifically, we consider five moments: their standard deviation and the first four autocorrelations. For each scenario (DT) and (ST), our goal is to determine a parameter set $\beta$ that, on average, brings these simulated moments as close as possible to their empirical counterparts. Closeness, in this context, is measured by a loss function $L(\cdot )$ that computes the mean quadratic distance between simulated and empirical moments across 10 simulation runs of (7) and (8). Averaging enhances the precision of parameter estimates compared to the ideal case where expected values of moments could be determined analytically; for comprehensive details and the improved precision achieved by 10 repetitions, see, e.g., Duffie and Singleton (Reference Duffie and Singleton1993, p. 945).

The objective function’s value depends on the parameter vector $\beta$ and a random seed, denoted by natural numbers $b$ , initializing the stochastic simulations; thus, $L = L({\beta }, b)$ in general. In total, $b = 1, 2, \ \ldots \, B = 100$ samples prove sufficient for the following two-stage procedure. In the first stage, for each $b$ , coefficients $\hat{\beta }^b$ are computed, which minimize the loss function over all admissible $\beta$ . Interestingly, in each case, all the empirical moments are (almost) perfectly matched, so that $L(\hat{\beta }^b, b) \approx 0$ for all $b$ . For the stochastic trend scenario, it should be added that this holds when the standard deviation ${\sigma }_{\varepsilon }$ of the random walk is fixed at values less than or equal to 0.0050. Any increase in ${\sigma }_{\varepsilon }$ beyond this threshold would progressively deteriorate the match. This finding led us to choose an upper limit of ${\sigma }_{\varepsilon } = 0.0050$ for the calibration. Within the framework of (ST) combined with (7), we consider this value as providing a maximal role to the randomness in the trend, one that still remains compatible with our current requirements for the first differences of GVA.

Having obtained the optimal parameters $\{\hat{\beta }^b\}_{b=1}^B$ in the first stage, it is crucial to consider that a single vector $\hat{\beta }^b$ is tailored to a particular random seed $b$ . Using $\hat{\beta }^b$ for simulations initialized with a different seed $c \neq b$ may generally lead to a suboptimal match, with $L(\hat{\beta }^b\!,c) \gt L(\hat{\beta }^b\!,b) \approx 0$ . To account for random variations, we opt for the random seed $b^o$ that yields the lowest mean loss across all realizations. The final parameter vector is therefore specified as:

(9) \begin{eqnarray}{\beta }^o \ = \ \hat{\beta }^{b^o} \, \, \, \, \textrm{where} \, \, \, \, b^o \ :=\, \, \mathop{\textrm{argmin}}\limits _b \, \Big \{\, \frac{1}{B} \, \,sum_{c=1}^B \ L(\hat{\beta }^b \!,\,c) \,\Big \} \end{eqnarray}

The outcome of this optimization procedure is reported in Table 2. The minimum losses $(1/B)\, \sum _c\, L({\beta }^o\!,c)$ in (9) show no significant difference between the deterministic and stochastic trend scenarios. Thus, both scenarios are suitable dynamic processes for generating artificial data in the subsequent section to test HRF and HP detrending in the context of GVA cyclical growth.

Table 2. Numerical coefficients ${\beta }^o$ (rounded) obtained from optimization (9)

5.1 Qualitative visual analysis

Expectations lean towards HRF being more effective in handling a random walk trend rather than a deterministic trend line. We, therefore, begin with a visual analysis of a typical example derived from (7) and (8) in the context of a stochastic trend (ST). In Figure 2, the The upper-left panel displays the data $y_t$ itself, depicted by the thin solid (black) line closely aligned with the thinner (blue) line representing its stochastic trend $y_t^\star$ . While these lines are challenging to differentiate due to their overall increase, the other three panels provide alternative perspectives. What can be better seen, however, is the main characteristic of the estimated trend $y_t^{\star,\textrm{HRF}}$ , the bold red line: it exhibits distinct swings, with the upper and lower turning points occurring slightly delayed compared to those in the data, evident at approximately $t = 5$ , $t = 33$ , $t = 37$ , and $t = 56$ .

Figure 2. HRF and HP were applied to a sample from (ST).

Because of the cyclical nature of $y_t$ , it is evident that this affects the resulting cyclical component $c_t^{\textrm{HRF}} = \,y_t - y_t^{\star,\textrm{HRF}}$ . As depicted in the upper-right panel of Figure 2, on these dates $c_t^{\textrm{HRF}}$ exhibits troughs and peaks before they actually occur in the true cyclical component $c_t = y_t - y_t^\star$ .Footnote ⁹ This phase shift is reminiscent of the lower-right panel in Figure 1, which was obtained in a more stylized setting by applying HRF to a strictly linear trend line and mildly disturbed regular sine waves around it. This deficiency is likely to be more general in nature, especially for growth series with a similar, relatively low noise level in their components.

The contrast between the HRF trend and the true random walk trend becomes more evident in the lower-left panel of Figure 2, which plots their difference from the deterministic trend (DT). While the relatively steady (blue) line illustrates the random walk’s gradual deviation from its deterministic counterpart, the HRF estimate captures this tendency with heightened volatility, mirroring the data series fluctuations that contribute to the observed phase shifts.

The random walk trend itself may appear relatively smooth in relation to the HRF trend, but nevertheless, its slope ranges between $-3$ and $+10$ percent per year, where the latter order of magnitude is not a rare exception. As another crude statistic, the standard deviation of the random walk trend increases to 2.35%, compared to 1.22% for the deterministic trend. These figures illustrate that there is indeed a substantial difference between the two frameworks of (DT) and (ST).

In summary, applying HRF to the current series does not yield the typical detrending outcome. The lower-right panel in Figure 2 illustrates that the conventional HP filter outperforms HRF in recovering the true cyclical pattern. Quantitatively, the contemporaneous correlation coefficient between $c_t^{\textrm{HP}}$ and $c_t$ is 0.903, which is significantly greater than the 0.413 for $c_t^{\textrm{HRF}}$ . However, HP exhibited a downward bias in overall variability, with a standard deviation ratio of 0.640 compared to HRF’s overestimation, resulting in a ratio of 1.289.

5.2 Monte Carlo simulation study

We can now evaluate HRF and the two HP filters more systematically through the quantitative experiments outlined at the beginning of this section. Simultaneously, we take the opportunity to broaden the perspective on the Hodrick–Prescott approach, motivated by occasional suggestions in the literature that the HP filter may excessively incorporate the cycle into the trend (Gordon, Reference Gordon2003, p. 218; Franke et al., Reference Franke, Flaschel and Proaño2006, p. 460, fn 10). Therefore, in addition to the conventional smoothing parameter $\lambda = 1600$ , we explore the performance of an alternative, higher value. To this end, we make use of the recommendations of Franke et al. (2023). Accordingly, in addition to HRF, we tested two versions of HP: HPc, denoted as the “conventional” $\lambda = 1600$ , and HPe, with “e” for “enhanced.” The alternative parameters, however, differ for (DT) and (CT). Table 3 reports the alternative parameters, which differ for (DT) and (ST). In both cases, they are substantially greater than $\lambda = 1600$ , thus reducing the flexibility of the estimated trend and potentially increasing the overall variability of the cyclical component.Footnote ¹⁰

To assess and contrast HRF, HPc, and HPe, a statistical measure must be utilized to quantify the disparity between the estimated and true trends or between the estimated and true cyclical components. This can be done along three dimensions. The first statistic is the distance between the series of the estimated trend $y_t^{\star, \text{est}}$ and the true trend $y_t^\star$ , which is most naturally specified by the root mean square deviation (RMSD). To mitigate the effects of HP’s end-of-sample bias, the first and last four quarters are discarded in the corresponding summations. For HRF, the first $h+p = 8+4 = 12$ unavailable values of the sample period are omitted from these sums. Given our samples of 241 quarters from $t = 0$ to $t = 60$ and two series $x_t$ , $z_t$ , their RMSD is defined as:

(10) \begin{align} \text{RMSD}(x, z):= \sqrt{\frac{1}{241-8} \sum _{q = 1+4}^{241-4} (x_{t(q)} - z_t)^2}, \quad t(q) := (q-1)/4 \end{align}

and correspondingly modified for HRF. To make the distance between $y_t^{\star, \text{est}}$ and $y_t^\star$ independent of the size of the variations in the cyclical component $c_t = y_t - y_t^\star$ , we scale $\text{RMSD}(y^{\star, \text{est}}, y^\star )$ by the latter’s standard deviation, which equals $\text{RMSD}(y, y^\star )$ . For a trend estimation $y_t^{\star, \text{est}}$ , we therefore define our distance measure as:

(11) \begin{align} d = d(y^{\star, \text{est}}, y^\star ; y) := \frac{\text{RMSD}(y^{\star, \text{est}}, y^\star )}{\text{RMSD}(y, y^\star )} \end{align}

Hence, $d$ measures the deviations of the estimated trend from the true trend in the percent of the cyclical component’s variability, making values of $d$ considerably below unity desirable. This normalization provides a better sense of the overall quality of the estimation. It may be noted that the distance between $y_t^{\star, \text{est}}$ and $y_t^\star$ is equivalent to the distance between the corresponding cyclical components $c_t^{\text{est}} = y_t - y_t^{\star, \text{est}}$ and $c_t = y_t - y_t^\star$ .

Table 3. Smoothing parameters $\lambda$ for HPc (conventional) and HPe (enhanced)

The other two measures, as mentioned earlier, include the correlation between the estimated and true cyclical component, denoted as $\text{Corr}(c^{\text{est}},c)$ , and the standard deviation of $c_t^{\text{est}}$ in relation to that of $c_t$ , $\frac{\text{std}(c^{\text{est}})}{\text{std}(c)}$ . The treatment of the beginning and end of the sample period remains consistent with the specification of the distance $d$ .

In the main experiment, we generate 1000 sample runs of (7), (8) along with the DT or ST scenario. For each of these runs, the three performance statistics resulting from HRF, HPc, and HPe are computed, with Table 4 presenting the outcome. All the statistics exhibit a fairly symmetric distribution; therefore, the median values are sufficient to be reported. To provide context for the differences between the three detrending procedures, we also include the standard deviation (“std”) of the distributions. The results of our Monte Carlo experiment can, therefore, be summarized as follows:

1. 1. Regarding the two measures of distance and correlation, the Hamilton regression filter (HRF) generally performs worse than conventional Hodrick–Prescott detrending (HPc).

2. 2. On the other hand, in terms of the estimated variability of the cyclical component, HPc has a disadvantage compared to HRF. However, the tendency of HRF to overestimate the true variability needs to be emphasized.

3. 3. HPc performs reasonably well in the first two measures, but its consistent underestimation of the cyclical component’s variability also raises concerns.

4. 4. The enhanced HP filter (HPe) with appropriately increased smoothing parameter values exhibits notable improvements. In particular, it outperforms HPc and, to a greater extent, HRF in the distance and correlation measures.

5. 5. HPe, however, still exhibits a slight downward bias in the standard deviation of the cyclical component. Despite this, its small magnitude allows for potential bias correction in practical applications, as detailed in Franke et al. (Reference Franke, Kukacka and Sacht2022, Section 5.3).

Table 4. Performance statistics for HRF, HPc, HPe from 1000 sample runs

In summary, we find that the observed differences between the filters are significant, assuming the acceptance of (i) the study’s focus on growing output series such as GDP or GVA, (ii) the experimental framework involving the decomposition into a predefined trend and cyclical component, and (iii) the defined task for detrending procedures to identify these series accurately. As highlighted throughout the text, the HP approach’s central ambition aligns with the task at hand, and while HRF may adopt a somewhat different methodological perspective, users of HRF often share practical and interpretative views of the HP approach.